Problem: $\dfrac{ 5v + 8w }{ 3 } = \dfrac{ 7v + 5x }{ -7 }$ Solve for $v$.
Solution: Multiply both sides by the left denominator. $\dfrac{ 5v + 8w }{ {3} } = \dfrac{ 7v + 5x }{ -7 }$ ${3} \cdot \dfrac{ 5v + 8w }{ {3} } = {3} \cdot \dfrac{ 7v + 5x }{ -7 }$ $5v + 8w = {3} \cdot \dfrac { 7v + 5x }{ -7 }$ Multiply both sides by the right denominator. $5v + 8w = 3 \cdot \dfrac{ 7v + 5x }{ -{7} }$ $-{7} \cdot \left( 5v + 8w \right) = -{7} \cdot 3 \cdot \dfrac{ 7v + 5x }{ -{7} }$ $-{7} \cdot \left( 5v + 8w \right) = 3 \cdot \left( 7v + 5x \right)$ Distribute both sides $-{7} \cdot \left( 5v + 8w \right) = {3} \cdot \left( 7v + 5x \right)$ $-{35}v - {56}w = {21}v + {15}x$ Combine $v$ terms on the left. $-{35v} - 56w = {21v} + 15x$ $-{56v} - 56w = 15x$ Move the $w$ term to the right. $-56v - {56w} = 15x$ $-56v = 15x + {56w}$ Isolate $v$ by dividing both sides by its coefficient. $-{56}v = 15x + 56w$ $v = \dfrac{ 15x + 56w }{ -{56} }$ Swap signs so the denominator isn't negative. $v = \dfrac{ -{15}x - {56}w }{ {56} }$